Coupled fixed point theorems for a generalized Meir–Keeler contraction in partially ordered metric spaces

Let XX be a non-empty set and F:X×X→XF:X×X→X be a given mapping. An element (x,y)∈X×X(x,y)∈X×X is said to be a coupled fixed point of the mapping FF if F(x,y)=xF(x,y)=x and F(y,x)=yF(y,x)=y. In this paper, we consider the case when XX is a complete metric space endowed with a partial order. We define generalized Meir–Keeler type functions and we prove some coupled fixed point theorems under a generalized Meir–Keeler contractive condition. Some applications of our obtained results are given. The presented theorems extend and complement the recent fixed point theorems due to Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379–1393].